What are Mathematical Models and why are A-level students taught about them?

Mathematical models are threaded throughout the A-level curriculum. You can't escape the applications of modelling in the course whether its:

• Trig modelling for a fairground ride

• Sequences and Series modelling for population growth

• Exponential modelling for a car's value over time

• Quadratic modelling for a rugby throw

• Mechanics modelling for SUVAT, Moments etc.

• Statistical modelling within probability distributions

• Differential Equations for rates of change problems (cooling, heating, growth etc)

THEY ARE EVERYWHERE!

But why? What is the rationale for this?

Well mathematical models underpin a lot of real world applications within technology, science, research and computing.

Often when students come across modelling in the course they feel it has no real relevance. The aim of this post is to show you how crucial modelling is in the real world and WHY its important to take it seriously in A-level Maths!

What is a Mathematical Model?

A mathematical model attempts to provide an equation (or sets of equations) that attempt to model (or simulate) to a real world problem.

Take an example of the height of the height of a sea wave- looking at the image below you can see that we could simulate its features fairly well by using a sine function.

Models are used for a variety of purposes- often to help predict an output given certain input variables. Mathematical models are used as there are many functions within Mathematics that approximate as close to the real world problem.

Another example of this is a quadratic regression model which is useful to fit a quadratic curve to raw data.

Regression models in the field of data science are very useful as they are used to predict outcomes given certain variables. A very simple regression model is called Linear Regression and is very similar to problems involved line of best fit back at GCSE level.

Simple Linear regression to predict house prices based on number of bedrooms

Mathematical models are used as they also help with interpreting the real world problems in an easy way. Say for the linear regression picture above we can generate the equation of the green line to help predict house price. This line equation will be of the form y=mx+ c. As a result we can interpret the gradient 'm' in context (per increase in bedroom number what is increase in house price) and the y intercept (for a house with 0 bedrooms what is the house price- not relevant!).

In essence we can see some benefits of Mathematical models:

• They seek to be as close as possible to the real life problem

• They incorporate functions that help to simulate the problem

• They can be used to predict outputs given inputs

• They are relatively easy to interpret.

However...

A lot of these models work on assumptions!

Assumptions are used a lot by mathematicians to make things less complex. An example of assumptions in your A level course include the modelling assumptions in Mechanics - smooth surface, object modelled as particle, light and inextensible string.

Without these assumptions then mathematical models would become too convoluted to even warrant any effort. In the higher domains of data science there are even stricter assumptions!

Here is the list of assumptions for linear regression:

• Linearity: The relationship should look like a straight line.

• Independence: Each data point should be separate and not depend on the next one.

• Homoscedasticity: The data’s “scatter” should stay roughly the same everywhere.

• Normality: The errors should form a bell curve.

• No multicollinearity: Predictors shouldn’t be copies of each other.

• No autocorrelation: Errors shouldn’t form time‑based patterns.

• Correct specification: Use the right variables and the right type of model.

As you can see models can get quite complicated!

Not only do models have assumptions but certain models are only valid for particular input criteria. For example, a quadratic model which seeks to simulate the throw of a rugby ball will have a restriction for only positive y values of its height and time.

Another aspect to remember during the course of this blog is that Models are not perfect! Models seek to replicate the real world problem as close as possible but are prone to errors- in this case being far from the actual output data. In this case we can compare model performance by computing the difference between the raw actual output and predicted. There are many metrics in the realm of data science that seek to interpret model performance, one of which is called MSE (Mean Squared Error) and it looks like this:

Formula for MSE- useful for evaluating linear regression models

These metrics are relatively easy to calculate and can be used to evaluate model performance with a high degree of interpretability.

So why bother with Modelling at A level?

Since the curriculum was revamped in 2017 there has been a larger emphasis on Modelling in the course. The drive to ensure that the mathematics you learn is linked to the real world is part of the reason but also to incorporate problem solving and comprehension skills.

On a purely higher level however, what you learn in A-level is really the stepping stones to modelling problems that exist and are trying to be solved in the real world by highly intelligent people in their field.

Once you start to contemplate that then you are opening up new worlds to explore!